I think I will say Big Big Train. They are very catchy and use strong melodic structures but when you listen to it closely you can feel all the complex time signature and the multi-layers harmonies. THey are one of the most brilliant band of their time.

There are many accessible prog bands....off the top of my head the Moody Blues ,Renaissance , Tull, and Kansas come to mind....as well as IQ and other neo prog bands.

Whilst IQ are one of the finest bands ever, they are not really what I would call very accessible. Their music is quite complex and the lyrics can be impenetrable at times.

Marillion, Pendragon, Camel, Mostly Autumn and Renaissance are all truly excellent bands which a newcomer to prog would not find too intimidating. There are many others, but none so fine as these.

I agree with the bands you named as being 'accessible' but I don't think IQ is a difficult band at all....some of their lyrics are a bit murky as to the meaning but many of their ballads are very accessible imo.

Accessible as in a bit more commercial sounding, but still maintaining a flexible amount of complexity in the music I'd have to look at Peter Gabriel's solo career being a good start. 'So' is a very accessible Prog album. NINE INCH NAILS are another. Boy, Trent Reznor will give you just about anything these days. Actually, his latest album 'hesitation Marks' is a very accessible album. Another choice would be the infamous 'Duke' album from Genesis. Ummm...for more guitar based sound with less keyboards would be Fates Warning's Insideout album. Lastly, OSI. OSI are extremely accessible and I would rank them as my top recommendation for this forum. The album 'FREE' is a great example of just how accessible Prog can be. ;)

Jazz and Symphonic music in general are pretty accessible and can be very proggy.

there is a lot of soul music that is pretty accessible and yet very proggy.Jackie Brown OST

there is a lot of modern music that you cant classify due to a strong genre mix which can be very progressive.The whole electro genre is extremely progressive at times. There are amazing remixes out there, full of all kinds of genres. And it is accessible because it consists out of very modern stuff which is very well known even if it is complex and very flexible and unstraight. The people love minimalism/reductionism, but they also love diversity.

and of course you have to ask for, accessible for whom?Which background of genre you have?A hard rock, a pop soul or a melodic rock background?

no background and no prog restriction:Peter GabrielThe BeatlesThe DoorsAirRadioheadDredgAnathalloBroken Social SceneAnnualsGrizzly BearMewMumHanne HukkelbergTalk TalkThe NotwistThe XXTV on the RadioRage Against the Machine

from a restricted prog and rock definition:IZZBig Big TrainRPWLMarillionKansas

so maybe you recognize how much this relates to the background you have.many bands are very accessible if you just play the right songs and if not, they are also not accessible.

A lot of Crossover and Symphonic stuff that has that pop inflection like Kansas, Alan Parsons, Marillion, Spock's Beard (though they have a lot of weird/"inaccessible" moments), and Big Big Train. And of course the heavier stuff like Rush, Riverside and Porcupine Tree. Yes and Genesis aren't that hard to get into, in my opinion.

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